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Theorem disjss2 4188
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )

Proof of Theorem disjss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3344 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2783 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 3135 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A
y  e.  B ) )
42, 3syl 16 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A y  e.  B
) )
54alimdv 1632 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( A. y E* x  e.  A
y  e.  C  ->  A. y E* x  e.  A y  e.  B
) )
6 df-disj 4186 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x  e.  A
y  e.  C )
7 df-disj 4186 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
85, 6, 73imtr4g 263 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1726   A.wral 2707   E*wrmo 2710    C_ wss 3322  Disj wdisj 4185
This theorem is referenced by:  disjeq2  4189  0disj  4208  uniioombllem2  19480  uniioombllem4  19483  usgreghash2spotv  28529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2712  df-rmo 2715  df-in 3329  df-ss 3336  df-disj 4186
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